What is good software for cleaning up old photos?

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Adobe Elements. This is a simpler version of their Photoshop product. You can buy it (a quick scheck on the web shows it costs around US$70), but it is often bundled with scanners, so check to see if you have a copy.

The Gimp. This is open-source software, so can be downloaded from the Gimp website free of charge.

Adobe Photoshop. This is the industry standard for image editing. It is more complicated, but also more powerful - and more expensive at over USD500. In Hong Kong there is the option to buy for home / education use and pay less than half that amoount.

If I was just starting out, I'd use Adobe Elements if it was included with my scanner, otherwise I'd use The Gimp.

If you're going to be spending a lot of time working on old photos, and are willing to invest time in learning the software, Adobe Photoshop is worth the investment.

Or you may need to switch to Photoshop if the other software doesn't do what you need. eg I used The Gimp for several years. But when I started getting more very faded photos to work with, I wanted to scan and edit in 48-bit mode. Neither Elements or The Gimp supported it, so I switched to Photoshop.